arXiv:hep-th/0007191v4 11 Sep 2000 uegaiyadaCnnn ag Theory: Gauge Confining a and Supergravity eomdcnfl.W oeas httesadr oe itse model cascade. standard duality the a that of also base note We conifold. deformed r nege eetdSiegdaiytasomtosi transformations Seiberg-duality repeated undergoes ory lzto fcia ymtybekn n ofieet esu We gives confinement. conifold and the breaking of N symmetry deformation chiral the of IR the alization the incorporating In hep-th/0002159, couplings. of of that approaches it UV li ysoigta h relevant dec the flux that 5-form showing by Its plain [hep-th/0002159]. conifold the on branes ihsalcraueeeyhr fte‘ of coupling Hooft ‘t the if everywhere curvature conifol small fiel the with the br deforming to symmetry dual by background chiral pure-supergravity its hep-th/0002159 non-singular confines; of theory singularity gauge the the IR the in Far agMlsmyb ult tig rpgtn tsmall at propagating strings to dual be may Yang-Mills 1 = ervsttesnua I uegaiyslto describi solution supergravity IIB singular the revisit We χ BRslto fNkdSingularities Naked of SB–Resolution b colo aua cecs nttt o dacdStudies Advanced for Institute Sciences, Natural of School grR Klebanov R. Igor a eateto hsc,PictnUniversity Princeton Physics, of Department ult Cascades Duality rneo,N 84,USA 08540, NJ Princeton, rneo,N 84,USA 08544, NJ Princeton, eray1 2008 1, February a N Abstract n ate .Strassler J. Matthew and and SUSY 1 = SU ( N + M which n hoyo l scales, all on theory d ) g × ess hc eex- we which reases, s M ng PUPT-1944 IASSNS–HEP–00/56 SU .W rps a propose We d. g emtia re- geometrical a fmylea the at lie may lf oaihi flow logarithmic s gs htpure that ggest M slre nthe In large. is aigremoves eaking M ( N N rcinl3- fractional nawarped a on ag the- gauge ) b → , N − M . 1 Introduction

A fruitful extension of the basic AdS/CFT correspondence [1, 2, 3] stems from study- ing branes at conical singularities [4, 5, 6, 7]. Consider, for instance, a stack of

D3-branes placed at the apex of a Ricci-flat 6-d cone Y6 whose base is a 5-d Einstein manifold X5. Comparing the metric with the D-brane description leads one to con- jecture that type IIB string theory on AdS X is dual to the low-energy limit of 5 × 5 the world volume theory on the D3-branes at the singularity. A useful example of this correspondence has been to study D3-branes on the coni- fold [6]. When the branes are placed at the singularity, the resulting = 1 supercon- N formal field theory has gauge group SU(N) SU(N). It contains chiral superfields × A1, A2 transforming as (N, N) and superfields B1, B2 transforming as (N, N), with superpotential = λǫijǫklTrA B A B . The two gauge couplings do not flow, and W i k j l indeed can be varied continuously without ruining conformal invariance.

For many singular spaces Y6 there are also fractional D3-branes which can exist only within the singularity [8, 9, 10, 11]. These fractional D3-branes are D5-branes wrapped over (collapsed) 2-cycles at the singularity. In the case of the conifold, the singularity is a point. The addition of M fractional branes at the singular point changes the gauge group to SU(N + M) SU(N); the four chiral superfields remain, × now in the representation (N + M, N) and its conjugate, as does the superpotential [10, 11]. The theory is no longer conformal. Instead, the relative gauge coupling 2 2 g− g− runs logarithmically, as pointed out in [11], where the supergravity equations 1 − 2 corresponding to this situation were solved to leading order in M/N. In [12] this solution was completed to all orders; the conifold suffers logarithmic warping, and the relative gauge coupling runs logarithmically at all scales. The D3-brane charge, i.e. the 5-form flux, decreases logarithmically as well. However, the logarithm in the solution is not cut off at small radius; the D3-brane charge eventually becomes negative and the metric becomes singular. In [12] it was conjectured that this solution corresponds to a flow in which the gauge group factors repeatedly drop in size by M units, until finally the gauge groups are perhaps SU(2M) SU(M) or simply SU(M). It was further suggested that the × strong dynamics of this gauge theory would resolve the naked singularity in the metric. Here, we show that this conjecture is correct. The flow is in fact an infinite series of Seiberg duality transformations — a “duality cascade” — in which the number of colors repeatedly drops by M units. Once the number of colors in the smaller gauge group is fewer than M, non-perturbative effects become essential. We will show that these gauge theories have an exact anomaly-free Z2M R-symmetry, which is broken

1 dynamically, as in pure = 1 Yang-Mills theory, to Z . In the supergravity, this N 2 occurs through the deformation of the conifold.1 In short, the resolution of the naked singularity found in [12] occurs through the chiral symmetry breaking of the gauge theory. The resulting space, a warped deformed conifold, is completely nonsingular and without a horizon, leading to confinement. If the low-energy gauge theory has fundamental matter, a horizon appears and leads to screening.

2 Branes and Fractional Branes on the Conifold

2.1 The Conifold

The conifold is described by the following equation in C4:

4 2 zn =0 . (1) nX=1 Equivalently, using z = 1 σn z , where σn are the Pauli matrices for n =1, 2, 3 ij √2 n ij n 4 and σ is i times the unit matrix,P it may be written as

det zij =0 . (2) i,j

This is a cone whose base is a coset space T 11 =(SU(2) SU(2))/U(1), with topology × S2 S3 and symmetry group SU(2) SU(2) U(1). As discussed in [10], fractional × × × D3 branes at the singularity zn = 0 of the conifold are simply D5-branes which are wrapped on the S2 of T 11. The Einstein metric of T 11 may be written down explicitly [14]: 2 2 2 2 1 1 2 2 2 ds 11 = dψ + cos θ dφ + dθ + sin θ dφ . (3) T 9 i i 6 i i i  Xi=1  Xi=1   It will be useful to employ the following basis of 1-forms on the compact space [15]:

e1 e3 e2 e4 g1 = − , g2 = − , √2 √2 e1 + e3 e2 + e4 g3 = , g4 = , √2 √2 g5 = e5 , (4)

1For a five-dimensional supergravity approach to chiral symmetry breaking, see [13].

2 where

e1 sin θ dφ , e2 dθ , ≡ − 1 1 ≡ 1 e3 cos ψ sin θ dφ sin ψdθ , ≡ 2 2 − 2 e4 sin ψ sin θ dφ + cos ψdθ , ≡ 2 2 2 e5 dψ + cos θ dφ + cos θ dφ . (5) ≡ 1 1 2 2 In terms of this basis, the Einstein metric on T 11 assumes the form

4 2 1 5 2 1 i 2 ds 11 = (g ) + (g ) , (6) T 9 6 Xi=1 and the metric on the conifold is

2 2 2 2 ds6 = dr + r dsT 11 . (7)

2.2 The Gauge Theory

If we place N D3-branes and M fractional D3-branes on the conifold, we obtain an SU(N + M) SU(N) gauge group. The two gauge group factors have holomor- × phic scales Λ1 and Λ˜ 1. The matter consists of two chiral superfields A1, A2 in the

(N + M, N) representation and two fields B1, B2 in the (N + M, N) representation. The superpotential of the model is

ik jℓ W = λ1tr (AiBjAkBℓ)ǫ ǫ . (8)

The model has a SU(2) SU(2) U(1) global symmetry; the first (second) factor × × iα rotates the flavor index of the Ai (Bi), while the “baryon” U(1) sends Ai Aie , iα 2 → B B e− . There are also two spurious U(1) transformations, one an R-symmetry i → i and one a simple axial symmetry, under which λ1,Λ1 and Λ˜ 1 are generally not invari- ant. The charges of the matter and the couplings under the symmetries (excepting the SU(2) flavor symmetries) are given in Table 1. Although U(1)A and U(1)R are anomalous, there is a discrete Z2M R-symmetry under which the theory is invariant. In particular, if we let

[A , B ] [A , B ]e2πin/4M , n =1, 2,..., 2M , (9) i j → i j and rotate the gluinos by e2πin/2M , then the superpotential rotates by e2πin/M with

λ1, Λ1 and Λ˜ 1 unchanged. 2The question of whether this U(1) is actually gauged is subtle. We believe that it is global, and arguments for this were given in [6, 7, 16].

3 SU(N+) SU(N) SU(2) SU(2) U(1)B U(1)A U(1)R

A , A N N 2 1 1 1 1 1 2 + 2N+N 2N+N 2

1 1 1 B1, B2 N+ N 1 2 − 2N+N 2N+N 2

3N+ 2N 2 Λ − 0 2M 1 N+

3N 2N+ 2 Λ˜ − 0 2M 1 N − 2 λ1 0 0 − N+N

Table 1. Quantum numbers in the SU(N + M) SU(N) model; we have written × N N + M for concision. + ≡

The classical field theory is well aware that it represents branes moving on a conifold [6, 7]. Let us consider the case where the Ai and Bk have diagonal expectation values, A = diag(a(1), , a(N)), B = diag(b(1), , b(N)). The F-term equations h ii i ··· i h ii i ··· i for a supersymmetric vacuum

B A B B A B =0 , A B A A B A =0 (10) 1 i 2 − 2 i 1 1 k 2 − 2 k 1 are automatically satisfied in this case, while the D-term equations require a(r) 2 + | 1 | a(r) 2 b(r) 2 b(r) 2 = 0. Along with the phases removed by the maximum abelian | 2 | −| 1 | −| 2 | subgroup of the gauge theory, the D-terms leave only 3N independent complex vari- (r) (r) (r) ables. Define nik = ai bk ; then the D-term and gauge invariance conditions are (r) satisfied by using the nik as coordinates. These 4N complex coordinates satisfy the condition, for each r, (r) det nik =0 . (11) i,k

(r) This is the same as equation (2). Thus, for each r = 1, , N, the coordinates n11 , (r) (r) (r) ··· n12 , n21 , n22 , are naturally thought of as the position of a D3-brane moving on a conifold. There are various combinations of the fields and parameters which are invariant

4 under the global symmetries. One is ˜ 3N 2(N+M) 3M Λ1 − ik jℓ 2M I1 λ1 3(N+M) 2N [tr (AiBjAkBℓǫ ǫ )] (12) ∼ Λ1 − In addition, there are simple invariants such as

ik jℓ (1) tr [AiBj]tr[AkBℓ]ǫ ǫ R1 = ik jℓ ; (13) tr (AiBjAkBℓǫ ǫ ) there are many other similar invariants, in each of which the same number of A and B fields appear in numerator and denominator but with color and flavor indices contracted differently. Finally there is a constant invariant

(N+M)+N 3(N+M) 2N 3N 2(N+M) J λ Λ − Λ˜ − (14) 1 ≡ 1 1 1 which plays the role of a dimensionless complex coupling analogous to τ in = 4 N Yang-Mills. The superpotential of the model will be renormalized and takes the general form

ik jℓ (s) W = λ1tr (AiBjAkBℓ)ǫ ǫ F1(I1,J1, R1 ) (15) where F1 is a function which we will not fully determine.

2.3 The conformal case: M =0

If there are no fractional D3-branes, then the U(1)R is anomaly-free, and the theory is superconformal (or if the couplings g1,g2,λ are chosen completely arbitrarily, it will flow until it becomes conformal in the infrared.) There are two dimensionless global invariants λ2Λ Λ˜ , the overall coupling τ +τ , and Λ˜ /Λ , the relative coupling τ τ , 1 1 1 2 1 1 1 − 2 which are built purely from the parameters and may be chosen arbitrarily. Thus [17, 18, 6] there are two exactly marginal operators in the theory which preserve the continuous global symmetries. (There are other marginal operators which partially preserve these symmetries.) This was the case studied in [6], where it was shown the supergravity dual of this field theory is simply AdS T 11. 5 × In order to match the two couplings to the moduli of the type IIB theory on AdS T 11, one notes that the integrals over the S2 of T 11 of the NS-NS and R-R 5 × 2-form potentials, B2 and C2, are moduli. In particular, the two gauge couplings are determined as follows [6, 7]: 1 1 φ 2 + 2 e− , (16) g1 g2 ∼

5 1 1 φ e− B2 1/2 , (17) 2 2 2 g1 − g2 ∼ ZS  −  3 where ( S2 B2) is normalized in such a way that its period is equal to 1. The matching betweenR the moduli is one of the simplest checks of the duality. It is further possible to build a detailed correspondence between various gauge invariant operators in the SU(N) SU(N) gauge theory and modes of the type IIB theory on AdS T 11 × 5 × [6, 19, 20]. In [21, 22], it was noted that there exists a type IIA construction [23] which is T-dual to N D3-branes at the conifold. It involves two NS5-branes: one oriented in the (12345) plane, and the other in the (12389) plane. The coordinate x6 is com- pactified on a circle of circumference l6, and there are N (1236) D4-branes wrapped around the circle. If the NS5-branes were parallel, then the low-energy field theory would be the = 2 supersymmetric SU(N) SU(N) gauge theory with bifunda- N × mental matter (this type IIA configuration is T-dual to N D3-branes at the = 2 N Z2 orbifold singularity). Turning on equal and opposite masses for the two adjoint chiral superfields corresponds to rotating one of the NS5-branes. Under this relevant deformation the Z orbifold field theory flows to the = 1 supersymmetric conifold 2 N field theory [6, 22]. In terms of the type IIA brane construction, the two gauge couplings are deter- mined by the positions of the NS5-branes along the x6 circle. If one of the NS5-branes is located at x6 = 0 and the other at x6 = a, then [22]

1 l6 a 1 a 2 = − , 2 = . (18) g1 gs g2 gs The couplings are equal when the NS5-branes are located diametrically opposite each other (in the type IIB language this corresponds to S2 B2 being equal to half of its period). As the NS5-branes approach each other, oneR of the couplings becomes strong. This simple geometrical picture will be useful for analyzing the RG flows in the following sections.

2.4 The RG cascade: M > 0

Now let us consider the effect of adding M fractional D3-branes, which as shown in [10] corresponds to wrapping M D5-branes over the S2 of T 11. The D5-branes serve as sources of the magnetic RR 3-form flux through the S3 of T 11. Therefore, the

3These equations are crucial for relating the SUGRA background to the field theory beta functions when the theory is generalized to SU(N + M) SU(N) [11, 12]. ×

6 supergravity dual of this field theory involves M units of the 3-form flux, in addition to N units of the 5-form flux:

F3 = M , F5 = N . (19) 3 11 ZS ZT In the SUGRA description the 3-form flux is the source of conformal symmetry break-

ing. Indeed, now B2 cannot be kept constant and acquires a radial dependence [11]:

φ B2 Me ln(r/r0) , (20) 2 ZS ∼

while the dilaton stays constant at least to linear order in M. Since the AdS5 radial coordinate r is dual to the RG scale [1, 2, 3], (17) implies a logarithmic running of 1 1 2 2 in the SU(N + M) SU(N) gauge theory. This is in accord with the exact g1 − g2 × β-functions:

d 8π2 2 3(N + M) 2N(1 γ) , (21) dlog(Λ/µ) g1 ∼ − − d 8π2 2 3N 2(N + M)(1 γ) , (22) dlog(Λ/µ) g2 ∼ − −

where γ is the anomalous dimension of operators TrAiBj. A priori, the conformal invariance of the field theory for M = 0 requires that γ = 1 + O(M/N). Taking − 2 the difference of the two equations in (21) we then find

8π2 8π2 2 2 M ln(Λ/µ)[3 + 2(1 γ)] , (23) g1 − g2 ∼ − in agreement with (20) found on the SUGRA side. The constancy of the dilaton φ to order M is consistent with the field theory only if γ = 1 + O[(M/N)2]. Fortunately, − 2 the field theory in Table 1 has an obvious symmetry M M, N N + M, which → − → to leading order in M/N is M M with N fixed. Clearly γ is even under this → − symmetry and so cannot depend on M/N at first order. The SUGRA analysis of [11] was carried out to the linear order in M/N. Luckily, it is possible to construct an exact solution taking into account the back-reaction of φ H3 and F3 on other fields [12]. In this solution e = gs is exactly constant, which 1 1 translates into the vanishing of the β-function for 2 + 2 in the dual field theory. As g1 g2 in [11], 4

F3 = Mω3 , B2 =3gsMω2 ln(r/r0) , (24) 1 H = dB =3g M dr ω , (25) 3 2 s r ∧ 2 4We are not keeping track of the overall factor multiplying M, which is determined by the flux quantization.

7 where 1 1 ω = (g1 g2 + g3 g4)= (sin θ dθ dφ sin θ dθ dφ ) , (26) 2 2 ∧ ∧ 2 1 1 ∧ 1 − 2 2 ∧ 2 1 ω = g5 (g1 g2 + g3 g4) . (27) 3 2 ∧ ∧ ∧ The relative factor of 3 in (24), which is related to the coefficients in the metric (3), appears to be related to the factor of 3 in the = 1 beta function (23). This gives N the correct value of beta function from a purely geometrical point of view.

Both ω2 and ω3 are closed. Note also that

g ⋆ F = H , g F = ⋆ H , (28) s 6 3 − 3 s 3 6 3 2 where ⋆6 is the Hodge dual with respect to the metric ds6. Thus, the complex 3-form G3 satisfies the self-duality condition i ⋆6 G3 = iG3 , G3 = F3 + H3 . (29) gs

This is consistent with G3 being either a (0, 3) form or a (2, 1) form on the conifold.

The Calabi-Yau form carries U(1)R charge equal to 2, while G3 does not transform under the U(1)R. Hence, the only consistent possibility appears to be that G3 is a harmonic (2, 1) form.5 It follows from (28) that 2 2 2 gs F3 = H3 , (30) µνλ which implies that the dilaton is constant, φ = 0. Since F3µνλH3 = 0, the RR scalar vanishes as well. The 10-d metric is

2 1/2 1/2 2 2 2 ds10 = h− (r)dxndxn + h (r)(dr + r dsT 11 ) , (31)

where g N + a(g M)2 ln(r/r )+ a(g M)2/4 h(r)= b +4π s s 0 s (32) 0 r4 and a is a constant of order 1. Note that, for the ansatz (31), the solution for h may be determined from the trace of the Einstein equation:6

3/2 2 2 2 2 2 2 h− h g F + H =2g F , (33) ∇6 ∼ s 3 3 s 3 5We are grateful to S. Gubser and E. Witten for discussions on this issue. 6 We are grateful to A. Tseytlin for explaining this to us.

8 2 2 2 6 3/2 where is the Laplacian on the conifold. Since F M r− h− , the solution (32) ∇6 3 ∼ follows directly. An important feature of this background, which is not visible to linear order in

M, is that F5 acquires a radial dependence [12]. This is because

F = dC + B F , (34) 5 4 2 ∧ 3 and ω ω vol(T 11). Thus, we may write 2 ∧ 3 ∼ F = + , = (r)vol(T11) , (35) 5 F5 ∗F5 F5 K and (r)= N + ag M 2 ln(r/r ) . (36) K s 0 The novel phenomenon in this solution is that the 5-form flux present at the UV scale r = r may completely disappear by the time we reach a scale r =r ˜ where (˜r)=0. 0 K This is related to the fact that the flux S2 B2 is not a periodic variable in the SUGRA solution: as this flux goes through a period, (r) (r) M which has the effect R K →K − of decreasing the 5-form flux by M units. We will shortly relate this decrease, which we refer to for now as the “RG cascade”, to Seiberg duality. In order to eliminate the asymptotically flat region for large r we use the well-

known device of setting b0 = 0 (this corresponds to choosing the special solution of sec. 5 in [12]). In terms of the scaler ˜, we then have 4πg (r)= ag M 2 ln(r/r˜) , h(r)= s [ (r)+ ag M 2/4] (37) K s r4 K s

This solution has a naked singularity at r = rs where h(rs) = 0. Writing L4 h(r)= ln(r/r ) , L2 g M , (38) r4 s ∼ s we then have a purely logarithmic RG cascade:

2 2 2 r L ln(r/rs) 2 2 2 ds = dxndxn + dr + L ln(r/rs)dsT 11 . (39) 2 q r2 L ln(r/rs) q q This is essentially the metric of sec. 5 in [12] expressed in terms of a different radial coordinate. Since T 11 expands slowly toward large r, the curvatures decrease there so that corrections to the SUGRA are negligible. Therefore, there is no obstacle for using this solution as r where the 5-form flux diverges. The field theory → ∞ explanation of the divergence is that the RG cascade goes on forever as the scale is increased, generating bigger and bigger N in the UV.

9 As the theory flows to the IR, the cascade must stop, however, because negative N is physically nonsensical. Thus, we should not be able to continue the solution (39) to r < r˜ where (r) is negative. The radius of T 11 at r =r ˜ is of order √g M. K s The gauge group at this scale is essentially SU(M), and it is satisfying to see the

appearance of gsM, which is the ‘t Hooft coupling. As usual, if the ‘t Hooft coupling is large then the SUGRA solution has small curvatures. Nevertheless, the fact that the solution of [12] is singular tells us that it has to be modified, at least in the IR. After understanding the RG cascade, we will study the dynamics of the corresponding field theory, and will see how this singularity is removed.

3 The =1 RG Cascade is a Duality Cascade N We now trace the jumps in the rank of the gauge group to a well-known phenomenon in the dual = 1 field theory, namely, Seiberg duality [24]. The essential observation 2N 2 is that 1/g1 and 1/g2 flow in opposite directions and, according to (21), there is a scale where the SU(N + M) coupling, g1, diverges. To continue past this infinite coupling, we perform a = 1 duality transformation on this gauge group factor. The SU(N + N M) gauge factor has 2N flavors in the fundamental representation. Under a Seiberg duality transformation, this becomes an SU(2N [N + M]) = SU(N M) gauge − − group with 2N flavors, which we may call ai and bi, along with “meson” bilinears

Mij = AiBj. The fields ai and bi are fundamentals and antifundamentals of SU(N), while the mesons are in the adjoint-plus-singlet of SU(N). The superpotential after the transformation 1 W = λ tr M M ǫikǫjℓF (I ,J , R(s))+ tr M a b , (40) 1 ij kℓ 1 1 1 1 µ ij i j

where µ is the matching scale for the duality transformation [17], shows the Mij are actually massive. We may integrate them out 1 0=2λ M ǫikǫjℓF (I ,J , R(s)) tr a b (41) 1 kℓ 1 1 1 1 − µ i j leaving a superpotential

ik jℓ (s) W = λ2tr aibjakbℓǫ ǫ F2(I2,J2, R2 ) (42)

Here F2, λ2, I2, J2 and R2 are defined similarly as in the original theory. Thus we obtain an SU(N) SU(N M) theory which resembles closely the theory we started × − with. 7 7 The fact that the quartic superpotential is left roughly invariant by the duality transformation in theories of this type has long been considered of interest. It was first noted in [17], where it was

10 Let us study the matching more carefully. We define, for reasons which will become clear in a moment, the strong coupling scale of the SU(N M) factor to be − Λ˜ 2. The strong coupling scale of the SU(N) factor is not the same as it was before the duality (since the number of flavors in the SU(N) gauge group has changed) and

its old scale Λ˜ 1 must be replaced with a new strong coupling scale Λ2. The matching conditions relating these scales are of the form 1 λ2 2 (43) ∝ µ λ1 and 3(N+M) 2N 3(N M) 2N 2N M 3N 2(N+M) M 3N 2(N M) Λ − Λ˜ − − µ λ Λ˜ − λ− Λ − − . (44) 1 2 ∝ ∝ 1 1 2 2 It is easy to check that I I and J 1/J . (45) 2 ∝ 1 2 ∝ 1 (Note that the inversion of J is a sign of electric-magnetic duality, the generalization of τ 1/τ.) Matching of baryon numbers in the Seiberg duality assures that (N→+M) − (N M) (s) (A) (a) − . We will not attempt to match the R . ∼ i With these matchings, the dual theory has the global charges given in Table 2. Remarkably, this theory has the same form as the previous one with N N → − M. Thus the renormalization group flow is self-similar: the next step is that the SU(N) gauge group now becomes strongly coupled, and under a Seiberg duality transformation the full gauge group becomes SU(N M) SU(N 2M), and so − × − forth.

SU(N) SU(N ) SU(2) SU(2) U(1)B U(1)A U(1)R −

1 1 1 a1, a2 N N 2 1 − 2NN− 2NN− 2

1 1 1 b1, b2 N N 1 2 − − 2NN− 2NN− 2

3N 2N− 2 − Λ2 0 N 2M

3N− 2N 2 Λ˜ − 0 2M 2 N− − λ 0 2 0 2 − NN−

used to study duality in SO(3) gauge theories, and in [18], where its wider significance in Seiberg duality transformations was established.

11 Table 2. Quantum numbers of the dual SU(N) SU(N M) theory; we have × − written N = N M for concision. − −

This flow will stop, of course, at or before the point where N kM becomes zero − or negative. Note that the Seiberg duality transformation is the same in both the so-called conformal window (3N > N > 3 N ) and in the free magnetic phase ( 3 c f 2 c 2 ≥ Nf > Nc+1). Even for Nf = Nc+1 the effect on the superpotential described in [25] is not essential, since it is accounted for in the function F . The first significant changes occur when N = N , since for N N the classical moduli space is drastically f c f ≤ c modified. Thus, the RG flow just described proceeds step by step until the gauge group has the form SU(M + p) SU(p), where 0

string duality but a motion which transforms one theory into its dual, one which a priori need not leave the infrared physics invariant. In particular, there is no direct relation between the classical brane motion, or semiclassical brane bending, and the actual dynamics of the field theories. To see that the IIA story gives the right answer, one should use its M theory generalization [27, 28, 29, 30], but even there, the M5-brane, about which only holomorphic information is available, does not generally match the dynamics of the field theory, which is not holomorphic. These issues have been studied and explained in detail; see especially [31]. We mention this only to warn our readers not to take this paragraph for more than a heuristic argument.

12 one segment and N in the other — exactly our starting point but with N N M. → − After the crossing, the NS5-branes are still bent in the same directions as before, so again their x6 positions become equal and we are led to repeat the motion around the circle. Finally, the number N becomes of order M and something more drastic should happen [33]. For this physics, the analysis of [28, 29, 30] becomes essential.

4 Chiral Symmetry Breaking and the Deformation of the Conifold

The solution of [12] is well-behaved for large r but becomes singular at sufficiently small r. The solution must be modified in such a way that this singularity is removed. In this section we argue that the conifold (2) should be replaced by the deformed conifold 4 2 2 z = 2 det zij = ǫ , (46) i − i,j Xi=1 in which the singularity of the conifold is removed through the blowing-up of the S3 of T 11. There are a number of arguments in favor of this idea. One suggestive observation is that in the solution of [12], the source of the singularity can be traced to the infinite 3 energy in the F3 field. At all radii there are M units of flux of F3 piercing the S of 11 3 2 T , and when the S shrinks to zero size this causes F3 to diverge. If instead the S3 remained of finite size, as occurs in the deformed conifold, this problem would be evaded. However, the most powerful argument that the conifold is deformed comes from the field theory analysis, which shows clearly that the spacetime geometry is modified by the strong dynamics of the infrared field theory. We will see that the theory has a deformed moduli space, with M independent branches, each of which has the shape

of a deformed conifold. The branches are permuted by the Z2M R-symmetry, which

is spontaneously broken down to Z2. This breaking of the R-symmetry is exactly what we would expect in a pure SU(M) = 1 Yang-Mills theory, although here it N proceeds through scalar as well as gluino expectation values. The theory will also have domain walls, confinement, magnetic screening, and other related phenomena. The complete analysis of the nonperturbative dynamics of the field theory in Table 1 is mathematically intensive, and we have not attempted it. In this section we present a simplified version of the analysis which captures the physics which we are interested in. In an appendix we present more general (although still partial) results

13 that show our conclusions are robust. In particular, our goal is to discover what happens in the far infrared of the flow, where the D3 brane charge has cascaded (nearly) to zero and only the M fractional D3 branes remain. If there are no D3 branes left, we expect we have pure = 1 N Yang-Mills in the far infrared, a theory which breaks its Z2M R-symmetry to Z2 and has M isolated vacua, domain walls, and confinement. However, while this may be correct, we have no access to the supergravity background through this analysis. What we need is a probe which can see if and how the fractional D3-branes have modified the conifold itself. The right choice, it turns out, is to probe the space with a single additional D3 brane. In this case the gauge group is SU(M + 1) “SU(1)” — in short, simply × SU(M + 1) — with fields Ci and Dj in the M + 1 and M + 1 representations, i, j = ik jl 1, 2, and with superpotential W = λCiDjCkDlǫ ǫ . Define Nij = CiDj, which is gauge invariant. As in the discussion surrounding equation (11), the expectation values of Nij specify the position of the probe brane; in the classical theory, we have deti,j Nij = 0, indicating the probe is moving on the original, singular conifold. At low energy the theory can be written in terms of these invariants and develops the nonperturbative superpotential first written down by Affleck, Dine and Seiberg [34]

1 3M+1 M−1 ik jℓ 2Λ WL = λNijNkℓǫ ǫ +(M 1) ik jℓ . (47) − "NijNkℓǫ ǫ #

The equations for a supersymmetric vacuum are

1 2Λ3M+1 M−1 0= λ ik jℓ M Nij . (48)  − "(NijNkℓǫ ǫ ) #    The apparent solution Nij = 0 for all i, j actually gives infinity on the right-hand side. The only solutions are then

3M+1 ik jℓ M 2Λ (NijNkℓǫ ǫ ) = M 1 . (49) λ −

ik jℓ As predicted, this equation has M independent branches, in each of which NijNkℓǫ ǫ th 3M+1 M 1 is a M root of Λ /λ − . The Z2M discrete non-anomalous R-symmetry rotates ik jℓ 2πi/M NijNkℓǫ ǫ by a phase e , and thus the M branches transform into one another

under the symmetry. In short, the Z2M is spontaneously broken down to Z2. The low-energy effective superpotential is

1/M W = Mλ N N ǫikǫjℓ M 2λΛ3M+1 (50) h ij kℓ i ∝ h i

14 which reflects the M branches. Most importantly, on each of these branches the classical condition on the Nij has been modified to read

3M+1 1/M 1 ik jℓ Λ det Nij = NijNkℓǫ ǫ = M 1 (51) i,j 2 [2λ] − !

Comparing with equation (46) we see that the probe brane in the quantum theory moves on the deformed conifold; the classical singularity at the origin of the moduli space has been resolved through chiral symmetry breaking.

The above constraint on the expectation values for Nij implies that in the pertur- bative region (where semiclassical analysis is valid) they can break the gauge group only down to SU(M), with no massless charged matter. This gauge theory is thus in the universality class of pure SU(M) Yang-Mills, and will share many of its qualita-

tive properties. However, the existence of massive matter Ci,Dj in the fundamental representation of SU(M) (note that if N is large then C ,D have mass λN ) im- h 11i 2 2 11 plies that confinement occurs only in an intermediate range of distances. As in QCD with heavy quarks, pair production of the massive quarks breaks the confining flux tubes, so a linear potential between external sources exists only between the length

scales 1/√T and mq/T , where T is the string tension and mq is the dynamical quark 1/2M 3M+1 M 1 mass. For N N Λ /λ − , their minimal values, we expect little h 11i∼h 22i ∼ sign of a linear potential at any length scale, as in physical QCD. Only for p =0 do we expect confinement at all scales. More generally, for 1

three adjoint chiral superfields (namely, three of the Nij) and is essentially a copy of = 4 Yang-Mills. Consequently, we expect no strong dynamics from the SU(p) N sector, and the theory is very close to SU(M + p) with 2p light flavors. In this case a similar analysis to the above is essentially correct. At large expectation values, the gauge theory is broken to SU(M) = 1 Yang-Mills times SU(p) = 4 Yang- N N Mills, with massive states in the bifundamental representation of the group factors. Details of this analysis are given in the appendix. As before, pair production of these massive states eliminates confinement at large distances; electric sources are screened by massive states which leave them charged only under the nonconfining group SU(p). The pattern of chiral symmetry breaking gives us another qualitative argument

why the conifold must be deformed. The original conifold has a U(1)R symmetry

under which the zij in (2) rotate by a phase. In Table 1 we saw this was broken by

15 instantons to Z2M , but for large M this is a 1/M effect and need not show up in the leading order supergravity. However, if we expect the infrared theory to behave sim- ilarly to pure = 1 Yang-Mills, then we expect this symmetry to be spontaneously N broken to Z2. This breaking is a leading-order effect and most definitely should be visible in the supergravity. The only natural modifications of the conifold are its resolution and its deformation; only the latter breaks the classical U(1)R symmetry, and it indeed breaks it to Z2, as is obvious from equation (46). As a final argument, we consider expectations from the IIA/M brane construction. Classically we have NS and NS’ branes filling four-dimensional space and extending in the v = x4 +ix5 and w = x8 +ix9 directions respectively. They are separated along 6 the compact direction x by a distance a, which along with l6 sets the two classical gauge couplings, as explained in (18). In one x6 segment between the NS and NS’ brane we suspend M + 1 D4 branes; in the other there is only one D4 brane. A single complete wrapped D4-brane — our probe — is free to move anywhere in the v,w,x7 space, independently of the other branes, while the other M suspended D4-branes are pinned to v = w = 0. To understand the quantum theory, we must move to M theory [28, 29, 30], where we combine x6 with the new compact coordinate x10 using 6 10 t = ex +ix . Classically the equations for the NS and NS’ brane are w =0, t = 1 and v =0, t = ea. The M theory expectation is that, in the quantum theory, the probe brane will become an independent M5 brane wrapped on the t directions, while the suspended D4-branes join with the NS and NS’ branes to make a single M5-brane, which we will refer to as our MQCD brane. This type of behavior was first seen in = 2 and N = 1 supersymmetric Yang-Mills [27, 28, 29, 30]. Indeed the MQCD brane which N appears in our case should be very similar to that of = 1 super-Yang-Mills, since N in the limit the x6 direction becomes large they should become equal. The brane for super-Yang-Mills fills the coordinates x0, x1, x2, x3 and is embedded in the coordinates 6 10 v = x4 + ix5,w = x8 + ix9, t = ex +ix as a Riemann surface defined through the equations M 2M M (vw) =ΛL , v = t . (52) Notice classically the equations include vw = 0, corresponding to the presence of the NS and NS’ brane. However, the quantum Yang-Mills M-brane has vw equal to a nonzero constant, and has M possible orientations, one for each possible phase of a condensate. What is the connection with the deformed conifold? As shown in [35], a type IIA NS-brane and NS’-brane satisfying the equation vw = 0, that is, intersecting at a point, are T-dual to the conifold. This lifts without change to M theory. We saw this

16 equation appears in the construction of classical Yang-Mills, and it will appear in our classical theory as well. Meanwhile, if the NS and NS’ branes are at the same t, that is, if they intersect, then they can be deformed into a single object with equation vw = constant = 0. This object is T-dual to the deformed conifold. Again this also 6 lifts without change to M theory. Now notice that the Yang-Mills M-brane has this as one of its defining equations (52). This shows the NS and NS’ brane have been glued together into a single object. Without the suspended D4-branes, this could only occur if the joined NS and NS’-brane had equal t coordinates, but in the presence of the suspended D4-branes, which extend along the t direction, the NS and NS’-branes can be separated in t, as in (52). Thus the Yang-Mills M-brane shows that the suspended D4-branes allow a quantum effect in M theory by which the conifold can be deformed even when the two gauge couplings (18) are both finite. In our case, we similarly expect the two Riemann surfaces — the probe and the MQCD brane — to have M branches, with a continuous variable specifying the position of the probe brane in the space, and a discrete variable labeling the orientation of the MQCD brane. However, when the probe is far away and the x6 direction is large, our MQCD brane should closely resemble that of Yang-Mills. We therefore expect the equations governing it to have the same qualitative form. In

particular, we expect that the Z2M discrete symmetry rotating the phase of t by 2π is M broken to Z2, through the modification of the equation vw =0to(vw) = constant. By T-duality this indicates that the classical conifold is quantum deformed by the fractional branes.

5 Back to Supergravity: The Deformed Conifold Ansatz

The field theory analysis of the previous section shows that the naive U(1) (really

Z2M ) R-symmetry is actually broken to a Z2. On the other hand, the SUGRA back- ground (31) has an exact U(1) symmetry realized as shifts of the angular coordinate ψ on T 11. The presence of this unwanted symmetry in the IR may also be the reason for the appearance of the naked singularity. In this section we propose that the solution of this problem is to replace the conifold by its deformation (46) in the ansatz (31). This indeed breaks the U(1) symmetry z eiαz , k = 1,..., 4, down to its Z subgroup z z . Another k → k 2 k → − k reason to focus on the deformed conifold is that it gives the correct moduli space for the field theory, as shown in the previous section.

17 The metric of the deformed conifold was discussed in some detail in [14, 15, 36]. It is diagonal in the basis (4):

2 1 4/3 1 2 5 2 2 τ 3 2 4 2 ds6 = ǫ K(τ) 3 (dτ +(g ) ) + cosh [(g ) +(g ) ] 2 " 3K (τ)  2  τ + sinh2 [(g1)2 +(g2)2] , (53)  2  # where (sinh(2τ) 2τ)1/3 K(τ)= − . (54) 21/3 sinh τ For large τ we may introduce another radial coordinate r via

r3 ǫ2eτ , (55) ∼ and in terms of this radial coordinate

2 2 2 2 ds dr + r ds 11 . (56) 6 → T The determinant of the metric (53) is

g ǫ8 sinh4 τ , (57) 6 ∼ which vanishes at τ = 0. Indeed, at τ = 0 the angular metric degenerates into 1 1 dΩ2 = ǫ4/3(2/3)1/3[ (g5)2 +(g3)2 +(g4)2] , (58) 3 2 2 which is the metric of a round S3 [14, 15]. The additional two directions, correspond- ing to the S2 fibered over the S3, shrink as 1 ǫ4/3(2/3)1/3τ 2[(g1)2 +(g2)2] . (59) 8

1/4 1 4/3 1/3 In what follows we will set ǫ = 12 , so that 2 ǫ (2/3) = 1. 2 3 The collapse of the S implies that at τ =0 F3 must lie within the remaining S ,

F (τ =0)= Mg5 g3 g4 , (60) 3 ∧ ∧

which may be shown to be a closed 3-form. On the other hand, for large τ, F3 should approach its value M g5 (g1 g2 + g3 g4) (61) 2 ∧ ∧ ∧

18 found in the UV ansatz (24). These two closed 3-forms differ by an exact one,

g5 (g1 g2 g3 g4)= d(g1 g3 + g2 g4) (62) ∧ ∧ − ∧ ∧ ∧ Therefore, the simplest ansatz which interpolates smoothly between τ = 0 and large τ is

F = M g5 g3 g4 + d[F (τ)(g1 g3 + g2 g4)] 3 ∧ ∧ ∧ ∧ 5 3 4 5n 1 2 1 3 2 4 o = M g g g (1 F )) + g g g F + F ′dτ (g g + g g ) , (63) ∧ ∧ − ∧ ∧ ∧ ∧ ∧ n o with F (0) = 0and F ( )=1/2. Note also that this ansatz preserves the Z symmetry ∞ 2 which interchanges (θ1,φ1) with (θ2,φ2).

A similarly Z2-symmetric ansatz for B2 is

B = g M[f(τ)g1 g2 + k(τ)g3 g4] . (64) 2 s ∧ ∧ Using the identity

g5 (g1 g3 + g2 g4)= d(g1 g2 g3 g4) , (65) ∧ ∧ ∧ − ∧ − ∧ we find that

1 2 3 4 1 5 1 3 2 4 H = dB = g M[dτ (f ′g g + k′g g )+ (k f)g (g g + g g )] . (66) 3 2 s ∧ ∧ ∧ 2 − ∧ ∧ ∧ We further have

= B F = g M 2ℓ(τ)g1 g2 g3 g4 g5 , (67) F5 2 ∧ 3 s ∧ ∧ ∧ ∧ where ℓ = f(1 F )+ kF . (68) − The most general radial ansatz for the 10-d metric, consistent with the symmetries of the deformed conifold, is

2 2 2 2 2 5 2 2 3 2 4 2 ds10 = A (τ)dxndxn + B (τ)(dτ )+ C (τ)(g ) + D (τ)[(g ) +(g ) ] +E2(τ)[(g1)2 +(g2)2] . (69)

The reason we are allowed to assume that A,...,E depend only on τ is that before the introduction of the 3-form fields, the metric has the form (69), and our ansatz for

F3 and H3 does not break this symmetry. The flux of F3 is distributed uniformly over the S3 near the apex of the deformed conifold; therefore, the M D5 branes wrapped over the S2 may be thought of as smeared over the S3.

19 µνλ It is not hard to check that F3µνλH3 = 0, which implies that the RR scalar vanishes. It is not a priori clear whether the dilaton is constant for the deformed solution, but in what follows we will assume that such a background does exist, i.e. that 2 2 2 gs F3 = H3 . (70) Furthermore, guided by the simple form of the solution constructed in [12] and re- viewed in section 2, we will assume that the 10-d metric takes the following form:

2 1/2 1/2 2 ds10 = h− (τ)dxndxn + h (τ)ds6 , (71)

2 where ds6 is the metric of the deformed conifold (53). This is the same type of “D- brane” ansatz as (31), but with the conifold replaced by the deformed conifold as the transverse space. This form will also permit additional D3-brane probes to be directly included in the ansatz. The type IIB equations satisfied by the 3-form fields are

φ φ 2 d(e ⋆ F )= F H , d(e− ⋆H )= g F F . (72) 3 5 ∧ 3 3 − s 5 ∧ 3 First, let us calculate ℓ(τ) ⋆ g M 2dx0 dx1 dx2 dx3 dτ . (73) F5 ∼ s ∧ ∧ ∧ ∧ K2h2 sinh2(τ)

To write down the first equation we need

1 0 1 2 3 2 τ 1 2 ⋆ F3 = Mh− dx dx dx dx (1 F ) tanh dτ g g ∧ ∧ ∧ ∧ " −  2  ∧ ∧

2 τ 3 4 5 1 3 2 4 +F coth dτ g g + F ′g (g g + g g ) (74).  2  ∧ ∧ ∧ ∧ ∧ # Assuming a constant φ and using (65) we find

2 2 d 1 ℓ (1 F ) tanh (τ/2) F coth (τ/2)+2h (h− F ′)= α(k f) , (75) − − dτ − K2h sinh2 τ 2 where α is a normalization factor proportional to (gsM) . Let us turn to the second of the equations (72). Since

1 0 1 2 3 5 2 τ 1 2 ⋆H3 = gsMh− dx dx dx dx g (k′ tanh g g − ∧ ∧ ∧ ∧ " ∧  2  ∧

2 τ 3 4 1 1 3 2 4 +f ′ coth g g ) (f k)dτ (g g + g g ) , (76)  2  ∧ − 2 − ∧ ∧ ∧ #

20 we find

d 1 2 1 ℓ(1 F ) h (h− coth (τ/2)f ′) (f k)= α − dτ − 2 − K2h sinh2 τ d 1 2 1 ℓF h (h− tanh (τ/2)k′)+ (f k)= α . (77) dτ 2 − K2h sinh2 τ where α is the same normalization factor as in (75). We have been assuming that the dilaton is constant. The equation that guarantees this is (70). Writing it out with our ansatz gives

2 2 2 (k′) (f ′) 2(f k) + + − cosh4(τ/2) sinh4(τ/2) sinh2 τ 2 2 2 (1 F ) F 8(F ′) = − + + . (78) cosh4(τ/2) sinh4(τ/2) sinh2 τ

In order to complete the system of equations we need the Einstein equations for the metric. In view of the simplified ansatz (71) for the metric it is sufficient to use the trace of the Einstein equation:

3/2 2 2 2 2 2 2 h− h g F + H =2g F , (79) ∇6 ∼ s 3 3 s 3 where now 2 is the Laplacian on the deformed conifold. Using (53), we find that ∇6 the explicit form of this equation is

2 2 2 1 d 2 2 α (1 F ) F 8(F ′) 2 (h′K (τ) sinh τ)= −4 + 4 + 2 . (80) sinh τ dτ − 4 " cosh (τ/2) sinh (τ/2) sinh τ #

5.1 The First-Order Equations and Their Solution

In searching for BPS saturated supergravity backgrounds, the second order equations should be replaced by a system of first-order ones (see, for instance, [37, 38]). Luckily, this is possible for our ansatz. We have been able to find a system of simple first-order equations, from which (75), (77), (78) and (79) follow:

2 f ′ = (1 F ) tanh (τ/2) , − 2 k′ = F coth (τ/2) , 1 F ′ = (k f) , (81) 2 − and f(1 F )+ kF h′ = α − . (82) − K2(τ) sinh2 τ

21 Note that the first three of these equations, (81), form a closed system and need to be solved first. In fact, these equations imply the self-duality of the complex 3-form 9 with respect to the metric of the deformed conifold: ⋆6G3 = iG3. Inspection of these equations shows that the small τ behavior is10

f τ 3 , k τ , F τ 2 . (83) ∼ ∼ ∼ On the other hand, for large τ the 3-forms have to match onto the conifold solution [12], τ τ 1 f , k , F . (84) → 2 → 2 → 2 Remarkably, it is possible to find the solution with these boundary conditions in closed form. Combining (81) we find the following second-order equation for F :

1 2 2 F ′′ = [F coth (τ/2)+(F 1) tanh (τ/2)] . (85) 2 − The solution is sinh τ τ F (τ)= − , (86) 2 sinh τ from which we obtain τ coth τ 1 f(τ) = − (cosh τ 1) , 2 sinh τ − τ coth τ 1 k(τ) = − (cosh τ + 1) . (87) 2 sinh τ Now that we have solved for the 3-forms on the deformed conifold, the warp factor may be determined by integrating (82). First we note that τ coth τ 1 ℓ(τ)= f(1 F )+ kF = − (sinh 2τ 2τ) . (88) − 4 sinh2 τ − This behaves as τ 3 for small τ. It follows that, for small τ,

2 h = a0 + a1τ + ... . (89)

For large τ we impose, as usual, the boundary condition that h vanishes. The resulting integral expression for h is

2/3 2 ∞ x coth x 1 1/3 h(τ)= α dx 2 − (sinh(2x) 2x) . (90) 4 Zτ sinh x − 9 We believe, according to a discussion in section 2.4, that G3 is a harmonic (2, 1) form on the deformed conifold. 10It is also possible to shift f and k by the same constant. The effect of this shift will be considered in section 5.2.

22 We have not succeeded in evaluating this in terms of elementary or well-known special functions. For our purposes it is enough to show that

3 1/3 4τ/3 h(τ 0) a ; h(τ ) 2 ατe− . (91) → → 0 →∞ → 4 This is nonsingular at the tip of the deformed conifold and, from (55), matches the form of the large-τ solution (38). The small τ behavior follows from the convergence 4x/3 of the integral (90), while at large τ the integrand becomes xe− . ∼ Thus, for small τ the ten-dimensional geometry is approximately R3,1 times the deformed conifold:

2 1/2 1/2 1 2 2 1 2 1 2 2 2 ds10 a0− dxndxn + a0 dτ + dΩ3 + τ [(g ) +(g ) ] . (92) →  2 4 

Very importantly, for large gsM the curvatures found in our solution are small everywhere. This is true even far in the IR. Indeed, since the integral (90) converges,

a α (g M)2 . (93) 0 ∼ ∼ s 3 Therefore, the radius-squared of the S at τ = 0 is of order gsM, which is the ‘t Hooft coupling of the gauge theory found far in the IR. As long as this is large, the curvatures are small and the SUGRA approximation is reliable. We have now seen that the deformation of the conifold allows the solution to be

non-singular. Qualitatively, this is because the conserved F3 flux prevents the 3-cycle from collapsing. This is why we expect to find a metric with a collapsing 2-cycle but finite 3-cycle, and these are the properties of the deformed conifold. It may be of further interest to consider more general metrics of the form (69), and to allow the dilaton to vary. In that event it still seems likely that the qualitative properties of the solution near the apex will not change.

5.2 Correspondence with the Gauge Theory

In this section we point out some interesting features of the SUGRA background we have found and show how they realize the expected phenomena in the dual field theory. In particular, we will now demonstrate that there is confinement and magnetic screening, and argue that there are domain walls and baryon vertices with a definite mass scale. In many ways our results resemble those found in the =1∗ theory [39], N but the specific details are quite different; the confining vacua of = 1∗ involve a N spacetime with a spherical 5-brane sitting in it, while our present spacetime is purely given by supergravity.

23 First we should ask the question: how does the dimensional transmutation mani- fest itself in supergravity? The answer is related to the presence of parameter ǫ in the deformed conifold metric (53). Reinstating this parameter is accomplished through

ds2 ǫ4/3ds2 . (94) 6 → 6 8/3 We are then free to redefine h hǫ− to remove the ǫ dependence from the trans- → verse part of the metric. Very importantly, the dependence then appears in the longitudinal part, and the metric assumes the form

2 1/2 2 1/2 2 ds10 = h− (τ)m dxndxn + h (τ)ds6 , (95)

so that m ǫ2/3 sets the 4-d mass scale. This scale then appears in all 4-d dimen- ∼ sionful quantities. Now let us see the theory has confining flux tubes. The key point is that in

the metric for small τ (92) the function multiplying dxndxn approaches a constant.

This should be contrasted with the AdS5 metric where this function vanishes at the horizon, or with the singular metric of [12] where it blows up. Consider a Wilson contour positioned at fixed τ, and calculate the expectation value of the Wilson loop using the prescription [41, 42]. The minimal area surface bounded by the contour bends towards smaller τ. If the contour has a very large area A, then most of the minimal surface will drift down into the region near τ = 0. From the fact that the

coefficient of dxndxn is finite at τ = 0, we find that a fundamental string with this surface will have a finite tension, and so the resulting Wilson loop satisfies the area

law. Since for large gsM the SUGRA description is reliable for all τ, we seem to have found a “pure supergravity proof” of confinement in = 1 gauge theory. A similar N result was found previously in [39] but involved a spacetime containing an NS5-brane with D3-brane charge. A simple estimate shows that the string tension scales as

m2 Ts . (96) ∼ gsM

To see that magnetic charge is screened, we must identify the correct massive magnetically-charged source. The correct choice is a fractional D1-brane, that is, a D3-brane wrapped on the S2 of T 11, attached to the boundary of the space at τ = . On the six-dimensional deformed conifold the S2 is fibered over τ such ∞ that the resulting three-dimensional bundle has only one boundary, at τ = ; near ∞ τ = 0 the S2 shrinks to zero size and the bundle locally has topology R3. Therefore, a D3-brane with a single boundary can be wrapped on this bundle, corresponding to a fractional D1-brane attached at τ = which quietly ends at τ = 0. Strictly ∞

24 speaking, this only shows monopole charge is not confined; to show it is screened one must go further and show this object does not couple to any massless modes.

As we showed in section 2, the field theory has an anomaly-free Z2M R-symmetry at all scales. The UV limit of our background, which coincides with the solution found in [12], has a U(1) R-symmetry associated with the rotations of the angular coordinate ψ. For large M it is is somewhat difficult to distinguish between the U(1) and its discrete subgroup Z2M . In fact, the anomaly in the U(1), which breaks it down to Z2M , is an effect of fractional D-instantons, the euclidean D-string world sheets propagating inside T 11. The Wess-Zumino term present in the D-string action, which is associated with the topologically non-trivial F3, has to be quantized (this is simply the F3 flux quantization). As a result, the phase in the D-string path integral assumes Z2M rather than U(1) values. Our metric provides a geometrical realization for the phenomenon of chiral sym- metry breaking found in the field theory; the dynamical breaking of the Z2M down to

Z2 occurs via the deformation of the conifold. In the pure supergravity limit we have discussed, the spontaneous chiral symmetry breaking generates an η′-like Goldstone boson (the zero mode in our solution corresponding to rotation of the coordinate ψ), which must get a mass of order 1/M from these fractional instantons. To see how this mass arises, and how it relates to the domain walls which we discuss in a moment, would be very interesting. It is by now clear why the conifold ansatz adopted in [12] and reviewed in section 2 is too restrictive: it has the U(1) symmetry everywhere. On the other hand, our

deformed conifold ansatz breaks it down to Z2, with the U(1) symmetry becoming asymptotically restored at large radius. Thus, the deformation of the conifold ties together several crucial IR effects: resolution of the naked singularity found in [12],

breaking of the chiral symmetry down to Z2, and quark confinement. At the same time, the deformation does not destroy the logarithmic running of the couplings found in [12] because it does not affect the geometry far in the UV. Due to the deformation, the full SUGRA background has a finite 3-cycle. We now interpret various branes wrapped over this 3-cycle in terms of the gauge theory. Note that the 3-cycle has the minimal volume near τ = 0, hence all the wrapped branes will be localized there. This should be contrasted with wrapped branes in AdS X where 5× 5 they are allowed to have an arbitrary radial coordinate. A wrapped D3-brane plays the role of a baryon vertex which ties together M fundamental strings. Note that for M = 0 the D3-brane wrapped on the S3 gave a dibaryon [10]; the connection between these two objects becomes clearer when one notes that for M > 0 the dibaryon has M uncontracted indices, and therefore joins M external charges. Meanwhile, a D5-

25 brane wrapped over the S3 appears to play the role of the domain wall separating two inequivalent vacua of the gauge theory. As we expect, flux tubes can end on this object [40], and baryons can dissolve in it; as in [39], we may also build the domain walls from the baryons. Indeed, D3 and D5-branes play the roles of baryon vertices

and domain walls in = 1∗; however in that case they do not wrap a cycle but N instead have a boundary on the NS5-brane in the space [39]. Calculations using the metric (95) show that the baryon mass is

M mM , (97) b ∼ while the D5-brane domain wall tension is

1 3 Twall m . (98) ∼ gs Additionally, one can obtain the glueball spectrum in this theory. To do so re- quires finding the spectrum of eigenmodes of various supergravity fields in the metric background we have constructed. Since the background is known explicitly as a func- tion of τ, the calculation should be no more difficult than in [43, 44]. Unlike the case

of =1∗, where the presence of a narrow throat near a single NS5-brane could make N the computation potentially unreliable for the lowest modes [39], there is no possi- ble subtlety here, as the bulk space is large and everywhere nonsingular. Of course, there will be Kaluza-Klein modes on the S3 which are not present in the pure =1 N Yang-Mills theory. These are analogous to the extra modes which appear in both [45] and [39]; their presence is expected, since they are necessary whenever pure = 1 N Yang-Mills is embedded into a theory that is fully in the supergravity regime. Only in the limit of pure = 1 Yang-Mills, which we discuss below, can they be removed. N A simple estimate of the glueball and KK modes masses shows that, in the SUGRA

limit both scale as m/(gsM). Comparing with the string tension, we see that

T g M(m )2 . (99) s ∼ s glueball Thus, there is a large separation of scales between string tension and glueball mass in supergravity (a similar problem was observed in [43, 44]) which goes away at small

gsM. Finally, we should address the possibility that N is not a multiple of M. Note that in our solution the 5-form flux vanishes for τ = 0:

3 F5 = ℓ(τ) τ . (100) Z ∼ This suggests that the IR solution given above describes a large number M of wrapped D5-branes without any D3-branes. Therefore, for small τ the background should be

26 dual to SU(M) gauge theory (the SUGRA is reliable only if both M and gsM are large). More generally, however, the field theory analysis tells us that theories that may arise in the IR have gauge groups SU(M + p) SU(p), with M >p 0. If M is × ≥ large and p is of order 1, then the dual supergravity background should be the same as for p = 0, to leading order in M. The extra p colors should come from p actual D3-branes, placed at various points in our background. Then the moduli space for each D3-brane is essentially the deformed conifold, in agreement with the field theory analysis. When far from τ = 0, the D3-branes represent the IR =4 SU(p) factor N in the theory. The ’t Hooft coupling on these branes is g p 1, so when they are s ≪ brought to τ = 0 the theory represented is essentially SU(M + p) with 2p classically massless flavors and a quartic superpotential. The nonperturbative analysis of this theory, given in section 4 and in the appendix, then applies, giving chiral symmetry breaking and a moduli space with M branches. Note that confinement is lost in the presence of the D3-branes, in agreement with the field theory. Strings hanging from the boundary can simply end on the D3- branes, corresponding to the statement that external sources are screened by massive dynamical quarks and end up carrying only SU(p) charge.11 The corresponding Wilson loop will have a perimeter law. Of course if the quarks are heavy (i.e., if the D3-branes are at large τ) then relatively short flux tubes should be stable. It would be interesting to actually demonstrate this fact, which follows not from topology but from quantum dynamics. On the other hand, if p is of the same order as M, then the flux due to the D3- branes is large and should be included in the SUGRA solution. First, let us try to change the boundary condition on F5 so that F5 no longer vanishes at τ = 0 but is p. We find a consistent solution for the 5-form by replacing ℓ(τ) ℓ(τ)+C, where ∼ 2 → C is a constant of order p/(gsM ). From (82) we find that the effect of this on the warp factor is h h + h˜ where → 1 h˜(τ)= αC ∞ dx . (101) 2 2 Zτ K (x) sinh x This yields a singular behavior of h˜ for small τ: αC h˜ . (102) ∼ τ 11Similar findings were also obtained in =1∗ [39]. Many of the =1∗ vacua have dynamical massive W -bosons, whose pair productionN eliminates confinement. TheN representation of this gauge theory physics in the string theory is closely related to the representation presented here and in the last paragraph of this section.

27 The new behavior of h does change significantly the physical interpretation of the 1/2 solution. Now the coefficient of the dxndxn term scales as τ for small τ; hence, the Wilson loop no longer satisfies the area law. Again, we find agreement with the field theory. This gravity background corresponds to making the charged matter as light as possible (that is, making the expectation values of the scalar fields all as small as possible.) In this regime we expect no metastable flux tubes; the dynamical charges in the fundamental representation of SU(M + p) will screen external electric sources, until the sources are charged only under SU(p), which does not confine. Thus, the new behavior of the metric (102) incorporates the loss of confinement found upon addition of dynamical quarks. However, supergravity may receive large corrections in the small τ region because curvatures blow up at τ = 0 where we find a 12 singular horizon. Thus, requiring that F5 does not vanish at τ = 0 actually causes a singularity. Can we construct a non-singular SUGRA solution which incorporates screening? We believe that the correct approach may be to add D3-brane sources with total charge p (this way F5 may smoothly turn on from zero at τ = 0 to p at some finite value of τ). This idea also agrees with the incorporation of small p via actual D3-branes. We postpone construction of such non-singular ‘Coulomb branch’ solutions until a later publication.

5.3 The Dual of Pure =1 Yang-Mills Theory N As we have shown above, supergravity serves as a reliable dual of a cascading SU(N + M) SU(M) gauge theory, provided that g M is very large. We have also shown × s that, under appropriate circumstances, at the bottom of the cascade, we essentially have an SU(M) theory, with the other gauge group disappearing. An immediate question that arises is: can our results be used to learn something about the pure glue = 1 theory? N To start answering this question, let us note that the field B2 is multiplied by gsM,

while the jumps in the cascade occur after B2 has changed by an amount of order 1. Thus, the range of τ which describes any particular gauge group in the cascade is of

order 1/(gsM). This implies the supergravity regime is not sufficient for constructing

such a dual, because for large gsM the cascade jumps occur very frequently, and we find the pure glue theory only for small τ. There, at the tip of the deformed conifold,

both B2 and F5 are very small, F3 is of order M, and the metric is approximately given by (92). To have a reliable dual of the pure glue theory, valid for a large range of τ, we

12 We are grateful to A. Tseytlin for useful discussions of this point.

28 need to take the limit of small gsM (and thus small B2, holding M fixed) which is the opposite of the limit where supergravity has small corrections. In this limit the

S3 at the apex of the conifold becomes small and the space acquires large curvature. This situation is familiar from previous studies aimed at finding a string theory dual of a pure glue gauge theory [45, 39]. Nevertheless, our work does constitute progress towards formulating a stringy dual because our SUGRA background captures the correct topology of the resulting string background. Indeed we are led to conjecture that the type IIB string dual of the pure glue = 1 SU(M) theory is given by a g M 0 limit of a warped N s → deformed conifold background, with M units of the F3 flux piercing its 3-cycle, and

with B2 and F5 approaching zero at the apex. This would be the space generated by the fractional D3-branes alone, with no admixture of regular D3-branes. Hence it is relevant to pure SU(M) theory with no quark flavors. Of course, studying such a

theory for small gsM is difficult due to the well-known problems with RR flux and large curvature. However, the self-dual 5-form flux, which brings in some additional problems, is small, which raises hopes of a novel sigma model formulation. We note also that the addition of a small number of D3-branes to this story will permit the study of the SU(M + p) SU(p) theory, which essentially reduces, for × small g and p M, to SU(M + p) with 2p flavors and an all-important quartic s ≪ superpotential. It is far from certain that this construction can give any insight into QCD, since the light charged scalars play such a central role in the dynamics. However, if these scalars can easily be removed (along with the gauginos) through explicit supersymmetry breaking, there might be additional interest in this approach.

6 Discussion

We have not addressed the question of how to compute field theory correlation func- tions in this context, where our space does not approach Anti-de-Sitter space at large r. However, it is easy to see this space still has a boundary, and from the behavior of h(τ) it is clear that the logarithm is a subleading effect at large r. Correspondingly, at large N M, there is a sense in which the operators in the field theory have the ≫ same spectrum that they have for M = 0, since γ 1 . We therefore believe that ≈ − 2 for low-lying supergravity modes, corresponding to operators of dimension much less than M, the story will not be modified in a significant way from that discussed in 3 [2, 3]. For operators of dimension ∆ > 2 M, we expect more interesting effects. These 3 operators appear to exist at scales where 2 N > ∆, but should be eliminated when 3 2 N < ∆. In the gauge theory, it is known what should occur [46]; operators of high

29 dimension present classically are actually removed by quantum effects, which in the low-energy dual theory appear as simple group theory. On the gravity side we may speculate that high-lying bulk modes propagate in from the boundary until the region 3 11 where N 2 ∆; there T has shrunk down such that these modes blow up into the ∼ 3 “giant gravitons” of [47]. Only modes with dimension ∆ < 2 M can propagate all the way to τ = 0. It is easy to see that our story of the duality cascade can be orientifolded. This is obvious from the type IIA string theory brane construction. It is also clear from the corresponding SO Sp gauge theory, although we have not analyzed the field × theory dynamics to see how the orientifolded conifold is deformed. A number of other modifications, including theories whose IIA version involves multiple NS and NS’ branes, could potentially be interesting. This might be especially true for theories which are qualitatively different in the infrared from pure Yang-Mills, such as those studied in [48]. Another interesting choice would be to orbifold the theory along the lines of [39], so that the low energy theory is non-supersymmetric SU(M)2 with a Dirac fermion in the bifundamental representation. In contrast to the case studied in [39], the masslessness of this fermion would be exact, as it is guaranteed by the Z2M R-symmetry, and therefore chiral symmetry breaking and confinement in this QCD-like theory could be exhibited in the supergravity regime. Finally, it is interesting to resurrect a scenario discarded five years ago for its apparent absurdity. Namely, it is conceivable that the standard model — a small gauge group — itself lies at the base of a duality cascade. This is certainly possible, since the addition of supersymmetry and some appropriately chosen massive matter at the TeV scale easily could make the theory into one which could emerge from such an RG flow. In [49] it was in fact pointed out that this was the natural scenario if the standard model, with its very small gauge groups, is a low-energy Seiberg- dual description of some other theory; every natural choice for an ultraviolet theory has a larger gauge group than the standard model, and typically hits a Landau pole below the Planck scale, requiring additional duality transformations, still larger gauge groups, more Landau poles, and continuation ad nauseum. This was termed the “duality wall” (since in some cases the duality transformations piled up so fast that an infinite number were required in a finite energy range.) But now we see this continuous generation of larger and larger gauge groups — ugly and unmotivated within field theory, and driving the field theory into highly non-perturbative regimes — can correspond to a perfectly natural spacetime background on which strings may propagate. If we imagine that the ultraviolet of the duality cascade is cut off in

30 a compact space (along the lines of [50], following [51]) we may conjecture that the standard model coupled to gravity is best described, at high energy, by a compactified string theory on a space with a logarithmic (or otherwise warped) throat, with the weakly coupled standard model emerging as a good description only at energies below, say, 1–100 TeV. Such a model provides another possible way, somewhat related to ideas of [50, 51, 52, 53, 54], to explain the hierarchy between the gravitational and cN/M electroweak scales: it is perhaps given by TeV= m e− , where M is of order P l × 2 to 5, c is a number of order one, and N is the number of colors of the gauge group at the Planck scale.

Acknowledgements

We are grateful to K. Dasgupta, S. Frolov, S. Gubser, S. Gukov, J. Maldacena, J. Polchinski, A. Tseytlin and E. Witten for useful discussions. The work of I.K. was supported in part by the NSF grant PHY-9802484 and by the James S. McDonnell Foundation Grant No. 91-48; that of M.J.S. was supported by NSF grant PHY95- 13835 and by the W.M. Keck Foundation.

7 Appendix

In this appendix we analyze the field theory in somewhat greater detail, confirming and extending the results of section 3. First, we may check the results of section 3 in another region of moduli space.

Consider first SU(M + 1) with two flavors. Suppose we permit C1 and D1 to have 2 equal expectation values v, so that N11 = v . This breaks the SU(M +1) to SU(M).

If λ were zero, this would leave SU(M) with one flavor C2 and D2, plus two gauge singlets C D = N and C D = N ; the corresponding strong coupling scale h 1i 2 12 2h 1i 21 would be ΛM+1/v2. However, the presence of nonzero λ gives mass to these fields, leaving the SU(M) gauge theory with a flavor of mass λv2. The effective Lagrangian

is then 1 M+1 2 M−1 2 Λ /v W =2λv N22 + (103) " N22 # ik jℓ which again leads to M branches with the correct values of NijNkℓǫ ǫ . That our discussion of the SU(M + 1) theory in section 3 was only part of the story can be seen by starting one step higher, with SU(2M +1) SU(M + 1), which × reduces after one duality transformation to the SU(M + 1) case. The SU(2M + 1)

31 gauge group has one more flavor than color, and therefore, as λ 0, the theory is 1 → governed by the results of [25]. For λ1 = 0 the superpotential must go over to

a det Pijb a b W 4M+1 CiaPijbDj , (104) → Λ1 − where a, b are color indices of SU(M + 1), and P AB, C A2M+1, D B2M+1. (1) ∼ ∼ ∼ From this we learn the function F1(I1,J1, R1 ) is not equal to one, and in fact, in the limit λ 0, that is, I ,J 0, we have F (I ,J , R(1)) I /J f(R(1)). The low 1 → 1 1 → 1 1 1 1 → 1 1 1 energy theory is then SU(M + 1) with two flavors Ci,Di butq with superpotential

ik jl W = λ2CiDjCkDlǫ ǫ F2(I2,J2) (105)

ik jl Here (CiDjCkDlǫ ǫ ) is the only invariant involving C and D; there are no R2 ratios. The low energy effective superpotential is now

ik jℓ WL = λNijNkℓǫ ǫ F2(I2,J2). (106)

Note that F (I 0,J 0) = 1+ I /J ; some other limits can be studied but 2 2 → 2 → 2 2 will not be needed here. The vacuumq equations are

∂F (I2,J2) 0= λ F (I2,J2)+ I2 Nij. (107) " ∂I2 # This gives an equation for I (N N ǫikǫjℓ)2M , whose solution must be 2 ∝ ij kℓ

I2 = G(J2) . (108)

The holomorphic function G(J2) is not zero everywhere (since for I2 0, J2 0 → ik →jℓ it is not zero) so it can only be zero at special points. Consequently NijNkℓǫ ǫ is ik jℓ 2πi/M generally nonzero. Since the Z2M symmetry rotates NijNkℓǫ ǫ by e , we again find M separate branches. Again there are no restrictions on the individual values of the Nij, so each branch takes the form of a deformed conifold, with a nonzero superpotential. Thus we obtain the same result as before; only the magnitude of the deformation is modified from our previous analysis. This analysis is too weak to rule out the possibility that there might be several independent solutions for I2 given a single value of J2. This would lead to several sets of branches, each set consisting of M copies of a deformed conifold; the different sets would have deformations of different magnitudes. In the limit Λ only one set 1 →∞ would remain, as in our earlier analysis of the SU(M + 1) theory. Next, we consider the case of SU(M+p) SU(p), 0

32 where α, β are SU(p) indices and a, b are SU(M + p) indices. If the SU(p) coupling were set to zero, then we would have an SU(M + p) gauge theory with 2p flavors. An Affleck-Dine-Seiberg superpotential would be generated, giving

1 3M+p M−p α β ik jℓ Λ1 W = λ(Nij)β (Nkℓ)αǫ ǫ +(M p) (109) − detij,αβ N ! where in the determinant we treat N as a 2p 2p matrix. A little algebra gives the × equations for a supersymmetric vacuum as

2 3M+p M α Λ1 det[(Nij)β ] M p (110) ∝ λ − ! and λ(N )α(N )β ǫikǫjℓ (λpΛ3M+p)1/M (111) ij β kℓ α ∝ It is possible to show that these equations represent M branches, each of which is p copies of the deformed conifold — in other words, the moduli space of p probe branes moving on the deformed conifold. First, note that N 0 (N )α is a gauge invariant ij ≡ ij α operator. If we demand that the SU(p)-adjoint fields (N )α 1 δα(N )γ vanish, then ij β − p β ij γ the equations above become

2 3M+p M 0 Λ1 det[(Nij)] M p (112) ∝ λ − ! and λN 0 N 0 ǫikǫjℓ (λpΛ3M+p)1/M (113) ij kℓ ∝ which gives M branches, each of which is a single copy of the deformed conifold. This region of moduli space corresponds to taking all p probe branes to have the same

positions on the conifold. As before the Z2M global symmetry is broken to Z2; it is easy to see that the superpotential on the M branches rotates by a phase under the broken Z . Expectation values for elements of (N )α 1 δα(N )γ correspond M ij β − p β ij γ to moving the p probe branes apart; taking the special cases where these fields are α diagonal, it is easy to show that each set of eigenvalues of (Nij)β , i, j = 1, 2, sweeps out its own copy of the deformed conifold. When the SU(p) gauge coupling is turned back on, the superpotential will include unknown functions of the invariants I, J, and R. These can be generated by a number of different physical phenomena, including instantons in regions where the SU(p) group is partially broken. However, as before, these functions change the quantitative features of the deformation of the conifold without altering the basic

33 picture we have obtained. Furthermore, we expect no additional significant infrared dynamics. Above the strong-dynamics scale for SU(M +p), the SU(p) gauge group is α infrared free. Below it, the SU(p) group contains three adjoint fields (Nij)β which have a trilinear superpotential — in short, a copy of = 4 Yang-Mills. The SU(p) sector N is therefore scale-invariant and nonconfining at low energy. Lastly, we expect that the SU(p) dynamics plays no role in the supergravity regime for p M. Supergravity ≪ requires we work at small gauge coupling and large ’t Hooft coupling for SU(M), but in this regime SU(p) will have small ’t Hooft coupling and will be described by weakly-coupled field theory. In the end, then, we again expect M branches, given by equations of the same qualitative form as above. The case p = M is the most subtle. For SU(2M) SU(M), the SU(2M) theory × has equal numbers of flavors and colors, and consequently its moduli space is modified quantum mechanically [25]. If we turn off the SU(M) coupling, the superpotential becomes W = λ(N )α(N )β ǫikǫjℓF (I /J )+ X(det[(N )α] ¯ Λ4M ) , (114) ij β kℓ α 1 1 1 ij β − BB − 2M where the “baryon” is the gauge invariant operator AM AM , and the anti-baryon is B 1 2 similarly constructed from Bi. Here the equations seem to have multiple solutions. One solution is X = 0 ; N = 0 ; = ¯ = iΛ2M . (115) B B 2M In this case, the SU(M) gauge group is unbroken and, when its coupling is restored, it generates M distinct and isolated vacua via usual gaugino condensation. Alterna- tively, we may have = ¯ = 0; det[(N )α]=Λ4M ; [(N )α(N )β ǫikǫjℓG (I /J )]M =Λ4M , (116) B B ij β 2M ij β kℓ α 1 1 1 2M where we have not determined G1. As before this leads to M branches, each of which has M probe branes moving on a deformed conifold. This suggests that the complete solution to a theory with gauge group SU(N + M) SU(N) might involve not one set of M branches but many. The smallest set × would consist of p N mod M D3-branes moving on the deformed conifold. The ≡ next smallest set would consist of p + M D3-branes. Next would follow a branch with p +2M D3-branes, and so forth, growing in size without limit. To see whether this is the case requires a more thorough and complete field theory analysis, which we have not performed. In any case, these partial results all support the main claims of the paper: that all branches which appear are consistent with probe branes moving on a deformed conifold, and that each branch is one of M identical branches which are rotated by the spontaneously broken Z2M R-symmetry.

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38